Let f:XYf \;\colon\; X \longrightarrow Y be a morphism in SeqSpec(Top cg)SeqSpec(Top_{cg}). Write

YjCone(f) Y \overset{j}{\longrightarrow} Cone(f)

for its mapping cone and

Path *(f)iX Path_\ast(f) \overset{i}{\longrightarrow} X

for its mapping cocone (def.) formed with respect to the standard cylinder spectrum from def. \ref{StandardCylinderSpectrumSequential}. Then the stable homotopy groups π \pi_\bullet (def. \ref{StableHomotopyGroups}) form commuting diagram of the form

π (Path *(f)) i * π (X) π (Y) π 1(Path *(f)) = = π +1(Cone(f)) π (X) f * π (Y) j * π (Cone(f)) , \array{ \cdots &\to& \pi_{\bullet}(Path_\ast(f)) &\overset{i_\ast}{\longrightarrow}& \pi_\bullet(X) &\overset{}{\longrightarrow}& \pi_\bullet(Y) &\longrightarrow& \pi_{\bullet-1}(Path_\ast(f)) &\to& \cdots \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{\simeq}} \\ \cdots &\to& \pi_{\bullet+1}(Cone(f)) &\longrightarrow& \pi_\bullet(X) &\overset{f_\ast}{\longrightarrow}& \pi_\bullet(Y) &\overset{j_\ast}{\longrightarrow}& \pi_\bullet(Cone(f)) &\to& \cdots } \,,

where the top and bottom are long exact sequences, and where the vertical isomorphisms are induced from the canonical morphism

Path *(f)Maps(S 1,Cone(f)) *. Path_\ast(f) \longrightarrow Maps(S^1, Cone(f))_\ast \,.

From the nature of the standard sequential cylinder spectrum X(I +)X \wedge (I_+) and the fact that limits and colimits of sequential spectra are computed degreewise (prop. \ref{LimitsAndColimitsOfSequentialSpectra}) it follows that the mapping cones and mapping cocones of sequential spectra are degreewise the mapping cones and mapping cocones of pointed topological spaces induced from the standard reduced cyclinder construction (def.) of pointed topological spaces.

Regarding the exactness of the top sequence:

The ordinary (unstable) homotopy groups of the component spaces form long exact sequences of homotopy groups to the left (exmpl.) yielding commuting diagrams of the form

π q+k+1(Y k) π q+k(Path *(f k)) π q+k(X k) π q+k(Y k) π q+k+2(Y k+1) π q+k+1(Path *(f k+1)) π q+k+1(X k+1) π q+k+1(Y k+1) π q+k+3(Y k+2) π q+k+2(Path *(f k+2)) π q+k+2(X k+2) π q+k+2(Y k+2) . \array{ \cdots &\to& \pi_{q+k+1}(Y_k) &\longrightarrow& \pi_{q+k}(Path_\ast(f_k)) &\longrightarrow& \pi_{q+k}(X_k) &\longrightarrow& \pi_{q+k}(Y_k) \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots &\to& \pi_{q+k+2}(Y_{k+1}) &\longrightarrow& \pi_{q+k+1}(Path_\ast(f_{k+1})) &\longrightarrow& \pi_{q+k+1}(X_{k+1}) &\longrightarrow& \pi_{q+k+1}(Y_{k+1}) \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots &\to& \pi_{q+k+3}(Y_{k+2}) &\longrightarrow& \pi_{q+k+2}(Path_\ast(f_{k+2})) &\longrightarrow& \pi_{q+k+2}(X_{k+2}) &\longrightarrow& \pi_{q+k+2}(Y_{k+2}) \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && \vdots && \vdots && \vdots && \vdots } \,.

Here the vertical morphisms are those entering the definition of stable homotopy groups (def. \ref{StableHomotopyGroups}), and one checks that these indeed make all the squares commute due to the respect of the component maps for the structure maps of the sequential spectra.

Now taking the colimit over the vertical morphisms yields the sequence

π (Path *(f)) i * π (X) π (Y) π 1(Path *(f)) \array{ \cdots &\to& \pi_{\bullet}(Path_\ast(f)) &\overset{i_\ast}{\longrightarrow}& \pi_\bullet(X) &\overset{}{\longrightarrow}& \pi_\bullet(Y) &\longrightarrow& \pi_{\bullet-1}(Path_\ast(f)) }

and that this is exact follows since on the category Ab of abelian group, forming filtered colimits is an exact functor (prop.).

Regarding the exactness of the top sequence:

Since the mapping cone of the mapping cone inclusion is the suspension, and since by example \ref{OmegaSigmaAdjunctionUnitOnSequentialSpectraIsStableWeakHomotopyEquivalence} there is an isomorphism

π (X)π +1(XS 1) \pi_\bullet(X) \simeq \pi_{\bullet+1}(X \wedge S^1)

it is sufficient to show that for every ff and qq the sequence

π q(X)f *π q(Y)j *π q(Cone(f)) \pi_q(X) \overset{f_\ast}{\longrightarrow} \pi_q(Y) \overset{j_\ast}{\longrightarrow} \pi_q(Cone(f))

is exact in the middle. It is clear from the construction of the mapping cone that the composite morphism is zero, therefore what remains to be shown is that every element in the kernel of j *j_\ast is in the image of f *f_\ast.

So let α| k:S k+qY k\alpha|_k \colon S^{k+q} \longrightarrow Y_k be a representative of an element απ q(Y)\alpha \in \pi_q(Y) such that S k+qα| kY kj kCone(f) kS^{k+q} \overset{\alpha|_k}{\longrightarrow} Y_k \overset{j_k}{\longrightarrow} Cone(f)_k is null-homotopic, in that there is a left homotopy (def.) η\eta of the form

S q+k * S q+k(I +) η Cone(f) k j k S q+k α| k Y k. \array{ S^{q+k} &\overset{}{\longrightarrow}& \ast \\ \downarrow && \downarrow \\ S^{q+k} \wedge (I_+) & \overset{\eta}{\longrightarrow}& Cone(f)_k \\ \uparrow && \uparrow^{\mathrlap{j_k}} \\ S^{q+k} &\underset{\alpha|_k}{\longrightarrow}& Y_k } \,.

In terms of the pushout of the top square, which is Cone(S q+k)Cone(S^{q+k}), this is equivalently a map η\eta' forming this commuting square:

S q+k Cone(S q+k) α| k η Y k j k Cone(f) k. \array{ S^{q+k} &\longrightarrow& Cone(S^{q+k}) \\ \downarrow^{\mathrlap{\alpha|_k}} && \downarrow^{\mathrlap{\eta'}} \\ Y_k &\underset{j_k}{\longrightarrow}& Cone(f)_k } \,.

Consider then the induced long cofiber sequences to the right (prop.), and paste a copy of the structure compatibility square for the homomrphism ff to it, to obtain a commuting diagram of the form

S q+k Cone(S q+k) S q+k+1 S q+k+1 α| k η κ α| kS 1 X k f k Y k j k Cone(f) k X kS 1 f kS 1 Y kS 1 σ k X σ k Y X k+1 f k+1 Y k+1. \array{ && S^{q+k} &\longrightarrow& Cone(S^{q+k}) &\longrightarrow& S^{q+k+1} &\overset{\simeq}{\longrightarrow}& S^{q+k+1} \\ && \downarrow^{\mathrlap{\alpha|_k}} && \downarrow^{\mathrlap{\eta'}} && \downarrow^{\mathrlap{\kappa}} && \downarrow^{\mathrlap{\alpha|_k \wedge S^1}} \\ X_k &\overset{f_k}{\longrightarrow}& Y_k &\underset{j_k}{\longrightarrow}& Cone(f)_k &\longrightarrow& X_k \wedge S^1 &\overset{f_k \wedge S^1}{\longrightarrow}& Y_k \wedge S^1 \\ && && && \downarrow^{\mathrlap{\sigma^X_k}} && \downarrow^{\mathrlap{\sigma^Y_k}} \\ && && && X_{k+1} &\overset{f_{k+1}}{\longrightarrow}& Y_{k+1} } \,.

Now the total vertical rectangle on the right appearing here, exhibits an element σ k Xκ\sigma^X_k \circ \kappa in π k+q+1(X k+1)\pi_{k+q+1}(X_{k+1}) which is a preimage under f *f_\ast of the original element α| k\alpha|_k exhibited at the successor stage α| k+1\alpha|_{k+1}: in terms of the directed sequence that defines the stable homotopy groups in def. \ref{StableHomotopyGroups}, we find

α| k α| k+1=f *(σ k Xκ) [S q+k,X k] * (S 1()) S q+k,Y k [S q+k+1,S 1X Y] * [S q+k+1,σ k Y] [S q+k+1,Y k+1] *. \array{ \alpha|_k && && \alpha|_{k+1} = f_\ast(\sigma^X_k \circ \kappa) \\ \in && && \in \\ [S^{q+k},X_k]_\ast &\overset{(S^1\wedge(-))_{S^{q+k},Y_k}}{\longrightarrow}& [S^{q+k+1}, S^1 \wedge X_Y]_\ast &\overset{[S^{q+k+1}, \sigma^Y_k]}{\longrightarrow}& [S^{q+k+1}, Y_{k+1}]_\ast } \,.

Finally regarding the vertical isomorphisms:

First notice that the vertical morphisms make commuting squares in the first place, by unwinding the limits which give the mapping cocones (def.) and the induced looping (def.):

* X * Y id Y j Cone(f) Maps(I +,Y) * id Maps(I +,Y) * Maps(I +,j) * Maps(I +,Cone(f)) * Y id Y j Cone(f) * * * lim Ωj: ΩY Path *(f) ΩCone(f). \array{ & \ast &\longrightarrow& X &\longrightarrow& \ast \\ & \downarrow && \downarrow && \downarrow \\ & Y &\overset{id}{\longrightarrow}& Y &\overset{j}{\longrightarrow}& Cone(f) \\ & \uparrow && \uparrow && \uparrow \\ & Maps(I_+,Y)_\ast &\overset{id}{\longrightarrow}& Maps(I_+,Y)_\ast &\overset{Maps(I_+,j)_\ast}{\longrightarrow}& Maps(I_+,Cone(f))_\ast \\ & \downarrow && \downarrow && \downarrow \\ & Y &\overset{id}{\longrightarrow}& Y &\overset{j}{\longrightarrow}& Cone(f) \\ & \uparrow && \uparrow && \uparrow \\ & \ast &\longrightarrow& \ast &\longrightarrow& \ast \\ {}^{\mathllap{\underset{\longrightarrow}{\lim}}}\downarrow \\ \Omega j \colon & \Omega Y &\longrightarrow& Path_\ast(f) &\longrightarrow& \Omega Cone(f) } \,.

Hence in summary we do have a commuting diagram of exact sequences as shown in the statement, and hence the fact that π (Path *(f))π +1(Cone(f))\pi_\bullet(Path_\ast(f))\to \pi_{\bullet+1}(Cone(f)) is an isomorphism follows from exactness via the five lemma.

Revised on May 25, 2016 13:32:28 by Urs Schreiber (